| L(s) = 1 | + 1.31·3-s − 5-s − 7-s − 1.28·9-s − 11-s − 2.68·13-s − 1.31·15-s − 4.68·17-s + 8.23·19-s − 1.31·21-s + 4.14·23-s + 25-s − 5.61·27-s + 5.05·29-s − 5.39·31-s − 1.31·33-s + 35-s + 4.76·37-s − 3.52·39-s − 1.16·41-s − 5.95·43-s + 1.28·45-s + 8.68·47-s + 49-s − 6.14·51-s − 0.769·53-s + 55-s + ⋯ |
| L(s) = 1 | + 0.756·3-s − 0.447·5-s − 0.377·7-s − 0.426·9-s − 0.301·11-s − 0.745·13-s − 0.338·15-s − 1.13·17-s + 1.88·19-s − 0.286·21-s + 0.864·23-s + 0.200·25-s − 1.08·27-s + 0.937·29-s − 0.969·31-s − 0.228·33-s + 0.169·35-s + 0.784·37-s − 0.564·39-s − 0.181·41-s − 0.907·43-s + 0.190·45-s + 1.26·47-s + 0.142·49-s − 0.860·51-s − 0.105·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.744135012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.744135012\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 1.31T + 3T^{2} \) |
| 13 | \( 1 + 2.68T + 13T^{2} \) |
| 17 | \( 1 + 4.68T + 17T^{2} \) |
| 19 | \( 1 - 8.23T + 19T^{2} \) |
| 23 | \( 1 - 4.14T + 23T^{2} \) |
| 29 | \( 1 - 5.05T + 29T^{2} \) |
| 31 | \( 1 + 5.39T + 31T^{2} \) |
| 37 | \( 1 - 4.76T + 37T^{2} \) |
| 41 | \( 1 + 1.16T + 41T^{2} \) |
| 43 | \( 1 + 5.95T + 43T^{2} \) |
| 47 | \( 1 - 8.68T + 47T^{2} \) |
| 53 | \( 1 + 0.769T + 53T^{2} \) |
| 59 | \( 1 + 4.02T + 59T^{2} \) |
| 61 | \( 1 + 0.407T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 2.19T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 8.51T + 79T^{2} \) |
| 83 | \( 1 - 9.90T + 83T^{2} \) |
| 89 | \( 1 + 9.80T + 89T^{2} \) |
| 97 | \( 1 - 8.85T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.050460202360305328337008957859, −7.37207485389399062554899644873, −6.92347361060044792832892053762, −5.87867326310156014659723813948, −5.13365815625629942101039169063, −4.40357677651608428477926045704, −3.34764858836314674056396974920, −2.94124237552311052838803183062, −2.07739778658422397951419670908, −0.63931551469346701881798071143,
0.63931551469346701881798071143, 2.07739778658422397951419670908, 2.94124237552311052838803183062, 3.34764858836314674056396974920, 4.40357677651608428477926045704, 5.13365815625629942101039169063, 5.87867326310156014659723813948, 6.92347361060044792832892053762, 7.37207485389399062554899644873, 8.050460202360305328337008957859
